From the facts that you reported shouldn't you have cleared the engine of fuel and then attempted a second start?
Yup!
I hadn't realized, though should have, that I had failed to clear fuel, and created a hazard for myself. My point is that you can put a lot more fuel into the airbox faster with several throttle pumps than several primer strokes. If I bungled it using the primer, I would have really got it wrong pumping the throttle for the same intended outcome.
And yes, you can really get a turbine start wrong. I have seen it from right seat! A few turbines I have flown have very strict procedures about clearing fuel after a failed start.